Geometry optimization
Fixed cell
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon in combination with a pseudodojo pseudopotential (see Pseudopotentials).
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[1, 1, 1])Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT and DFTK's AtomsBase integration to setup the hydrogen molecule. We employ a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using Unitful
using UnitfulAtomic
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
cell_vectors = [[a, 0, 0]u"bohr", [0, a, 0]u"bohr", [0, 0, a]u"bohr"]
h2_crude = periodic_system([:H => [0, 0, 0.]u"bohr",
:H => [r0, 0, 0]u"bohr"],
cell_vectors)FlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110642959254 -0.82 0.80 6.0 3.33s
2 -1.117204542721 -2.18 -1.87 0.80 1.0 1.76s
3 -1.117250539853 -4.34 -2.74 0.80 1.0 16.8ms
4 -1.117251167079 -6.20 -3.58 0.80 1.0 16.3ms
5 -1.117251187979 -7.68 -3.99 0.80 1.0 19.1ms
6 -1.117251190063 -8.68 -4.87 0.80 1.0 66.3ms
7 -1.117251190142 -10.11 -5.51 0.80 2.0 18.1ms
8 -1.117251190143 -11.83 -6.58 0.80 1.0 16.8ms
9 -1.117251190143 -13.40 -7.12 0.80 2.0 28.2ms
10 -1.117251190143 -15.05 -7.60 0.80 1.0 16.6ms
11 -1.117251190143 -14.88 -8.51 0.80 1.0 16.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.53 0.80 1.0 3.10s
2 -1.117251190143 -15.05 -10.17 0.80 1.0 16.3ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 16.0s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517993350 -2.46 0.80 1.0 67.9ms
2 -1.117520586012 -5.59 -3.49 0.80 1.0 15.8ms
3 -1.117520612995 -7.57 -4.23 0.80 1.0 15.7ms
4 -1.117520613449 -9.34 -5.15 0.80 1.0 16.6ms
5 -1.117520613464 -10.81 -5.93 0.80 1.0 35.3ms
6 -1.117520613465 -12.14 -6.44 0.80 1.0 16.3ms
7 -1.117520613465 -13.37 -7.46 0.80 1.0 16.2ms
8 -1.117520613465 + -14.81 -8.28 0.80 1.0 16.5ms
9 -1.117520613465 -14.48 -8.96 0.80 1.0 18.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264645 -1.69 0.80 1.0 10.6ms
2 -1.118339043937 -4.04 -2.70 0.80 1.0 20.9ms
3 -1.118340011115 -6.01 -3.44 0.80 1.0 15.4ms
4 -1.118340028555 -7.76 -4.36 0.80 1.0 15.3ms
5 -1.118340029139 -9.23 -5.12 0.80 1.0 21.0ms
6 -1.118340029166 -10.57 -5.62 0.80 1.0 16.7ms
7 -1.118340029168 -11.73 -6.68 0.80 1.0 18.0ms
8 -1.118340029168 -13.55 -7.58 0.80 1.0 15.8ms
9 -1.118340029168 -14.88 -8.12 0.80 1.0 17.7ms
10 -1.118340029168 + -14.88 -8.55 0.80 1.0 15.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029168 -9.05 0.80 1.0 10.6ms
2 -1.118340029168 -15.65 -10.36 0.80 1.0 79.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029168 -11.05 0.80 1.0 315ms
2 -1.118340029168 -15.35 -11.42 0.80 1.0 17.0ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029168 │ │ 0.00297055 │ 2.54s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721508 -3.09 0.80 1.0 12.5ms
2 -1.118346864759 -6.84 -4.11 0.80 1.0 16.6ms
3 -1.118346866237 -8.83 -4.86 0.80 1.0 15.7ms
4 -1.118346866262 -10.59 -5.78 0.80 1.0 15.9ms
5 -1.118346866263 -12.09 -6.55 0.80 1.0 16.3ms
6 -1.118346866263 -13.42 -7.04 0.80 1.0 16.3ms
7 -1.118346866263 + -15.05 -8.13 0.80 1.0 16.6ms
8 -1.118346866263 -14.88 -8.95 0.80 1.0 18.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 12.0ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 17.4ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 16.8ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 26.5ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 16.1ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 16.3ms
7 -1.118356101386 -13.59 -7.65 0.80 1.0 18.1ms
8 -1.118356101386 -14.51 -8.45 0.80 1.0 16.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 12.9ms
2 -1.118356101386 + -14.75 -9.82 0.80 1.0 25.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 12.8ms
2 -1.118356101386 + -14.27 -11.46 0.80 1.0 18.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 451ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 11.8ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 27.3ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 16.4ms
4 -1.118356102877 -14.15 -7.62 0.80 1.0 16.2ms
5 -1.118356102877 + -15.18 -8.40 0.80 1.0 15.8ms
6 -1.118356102877 + -15.35 -8.89 0.80 1.0 23.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 12.3ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 17.5ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 20.3ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 17.3ms
5 -1.118356105120 -15.35 -7.86 0.80 1.0 17.6ms
6 -1.118356105120 + -15.05 -8.35 0.80 1.0 19.0ms
7 -1.118356105120 -14.88 -9.45 0.80 1.0 16.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 10.7ms
2 -1.118356105120 -14.88 -11.02 0.80 1.0 15.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 10.5ms
2 -1.118356105120 + -14.18 -12.44 0.80 1.0 17.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.08299e-8 │ 405ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431829, -4.25512e-35, -1.73728e-34]u"a₀")
Atom(H, [ 1.44318, -2.6504e-35, 2.302e-34]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.486365865339029 a₀
Our result (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Variable cell
Recent versions of GeometryOptimization support cell optimization as well by passing variablecell=true to minimize_energy!.
For a plane-wave code like DFTK, variable cells pose an additional challenge: changes to the cell's size affect the used plane waves, leading to discontinuities in the energy and stresses. This can cause the geometry optimization to struggle and/or fail to converge.
A practical strategy to overcome this problem is Energy cutoff smearing. As a demonstration let us find the optimal lattice constant of silicon.
Like before we define a calculator, this time with a kinetic_blowup set to use energy cutoff smearing:
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials,
kinetic_blowup=BlowupCHV()),
basis_kwargs = (; kgrid=[2, 2, 2], Ecut=10)
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[2, 2, 2])And here is our starting silicon structure:
a = 10.0u"bohr" # Approximate Silicon lattice constant
cell_vectors = a/2 * [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
initial_silicon = periodic_system([:Si => ones(3)/8,
:Si => -ones(3)/8],
cell_vectors;
fractional=true)FlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5 5;
5 0 5;
5 5 0]u"a₀"
Atom(Si, [ 1.25, 1.25, 1.25]u"a₀")
Atom(Si, [ -1.25, -1.25, -1.25]u"a₀")
We now minimize, passing variablecell=true:
using GeometryOptimization
results = minimize_energy!(initial_silicon, calc; variablecell=true,
tol_virial=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.335242530153 -0.80 0.80 7.3 509ms
2 -8.340942254396 -2.24 -1.69 0.80 1.3 398ms
3 -8.341207657881 -3.58 -2.79 0.80 3.0 31.8ms
4 -8.341214247001 -5.18 -3.33 0.80 3.0 33.1ms
5 -8.341214270117 -7.64 -4.09 0.80 1.3 23.9ms
6 -8.341214272491 -8.62 -5.14 0.80 2.3 27.5ms
7 -8.341214272516 -10.60 -5.62 0.80 2.7 49.4ms
8 -8.341214272518 -11.76 -6.72 0.80 2.3 25.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.45 0.80 1.0 417ms
2 -8.341214272518 -14.27 -8.47 0.80 1.0 22.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.73 0.80 1.0 20.0ms
2 -8.341214272518 + -Inf -9.02 0.80 1.0 22.9ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ 2.84656e-33 │ 0.163458 │ -0.16 │ 20.1s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350369332939 -1.20 0.80 9.0 58.2ms
2 -8.351862177571 -2.83 -1.88 0.80 1.3 23.3ms
3 -8.351925928677 -4.20 -2.78 0.80 3.0 31.6ms
4 -8.351928512530 -5.59 -3.71 0.80 3.3 81.9ms
5 -8.351928514689 -8.67 -5.14 0.80 2.7 29.1ms
6 -8.351928514755 -10.18 -5.87 0.80 4.7 35.2ms
7 -8.351928514756 -12.12 -6.55 0.80 2.7 27.7ms
8 -8.351928514756 -14.27 -7.69 0.80 2.3 26.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353372976300 -1.52 0.80 8.3 46.8ms
2 -8.353724442993 -3.45 -2.21 0.80 1.0 19.9ms
3 -8.353757345416 -4.48 -2.95 0.80 3.7 30.2ms
4 -8.353757519177 -6.76 -4.16 0.80 2.3 24.8ms
5 -8.353757531616 -7.91 -5.17 0.80 4.0 63.2ms
6 -8.353757531645 -10.54 -5.73 0.80 3.3 34.2ms
7 -8.353757531646 -11.85 -7.59 0.80 3.0 31.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531646 -8.20 0.80 1.0 18.8ms
2 -8.353757531646 -14.27 -8.28 0.80 1.0 20.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531646 -8.84 0.80 1.0 17.0ms
2 -8.353757531646 + -Inf -8.94 0.80 1.0 26.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531646 -9.46 0.80 1.0 18.2ms
2 -8.353757531646 + -14.45 -9.48 0.80 1.0 19.6ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531646 │ │ │ 0.0156413 │ -0.016 │ 3.83s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353684224601 -3.18 0.80 8.0 48.5ms
2 -8.353753068707 -4.16 -3.82 0.80 1.0 22.9ms
3 -8.353770530357 -4.76 -3.72 0.80 6.0 48.2ms
4 -8.353770536198 -8.23 -4.04 0.80 1.0 38.6ms
5 -8.353770536792 -9.23 -4.35 0.80 2.0 26.5ms
6 -8.353770537301 -9.29 -5.03 0.80 1.3 24.5ms
7 -8.353770537329 -10.55 -6.88 0.80 2.3 29.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353760771724 -2.00 0.80 8.3 58.9ms
2 -8.353871949822 -3.95 -2.71 0.80 1.0 21.6ms
3 -8.353895052234 -4.64 -3.30 0.80 4.3 33.2ms
4 -8.353895103929 -7.29 -4.12 0.80 2.0 22.9ms
5 -8.353895110675 -8.17 -5.01 0.80 3.0 31.8ms
6 -8.353895110750 -10.13 -5.57 0.80 3.7 32.7ms
7 -8.353895110753 -11.50 -6.91 0.80 1.7 23.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110753 -7.35 0.80 1.0 19.2ms
2 -8.353895110753 -13.75 -7.88 0.80 1.0 22.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110753 -8.38 0.80 1.0 24.3ms
2 -8.353895110753 -14.75 -8.59 0.80 1.0 20.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110753 -8.91 0.80 1.0 24.8ms
2 -8.353895110753 + -Inf -9.12 0.80 1.0 22.1ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110753 │ -3.86 │ 1.07896e-32 │ 0.000646203 │ -0.00065 │ 2.14s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353818202868 -3.47 0.80 8.3 47.3ms
2 -8.353881003717 -4.20 -3.77 0.80 1.0 22.7ms
3 -8.353895211924 -4.85 -3.81 0.80 6.0 51.5ms
4 -8.353895212165 -9.62 -4.00 0.80 1.0 22.8ms
5 -8.353895212437 -9.57 -4.35 0.80 1.0 22.7ms
6 -8.353895212487 -10.30 -4.54 0.80 1.0 22.6ms
7 -8.353895212503 -10.82 -5.08 0.80 1.0 26.6ms
8 -8.353895212506 -11.53 -5.74 0.80 2.0 27.0ms
9 -8.353895212506 -12.44 -6.55 0.80 2.7 28.1ms
10 -8.353895212506 -13.37 -6.88 0.80 3.0 32.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353821991792 -3.38 0.80 8.3 46.5ms
2 -8.353884740932 -4.20 -3.90 0.80 1.0 24.6ms
3 -8.353895362545 -4.97 -4.24 0.80 6.0 40.2ms
4 -8.353895362644 -10.00 -4.94 0.80 1.3 20.6ms
5 -8.353895362684 -10.41 -5.41 0.80 2.3 25.2ms
6 -8.353895362686 -11.70 -5.87 0.80 2.0 29.1ms
7 -8.353895362686 -12.79 -7.09 0.80 1.7 22.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.27 0.80 1.0 17.9ms
2 -8.353895362686 -14.05 -8.22 0.80 1.0 25.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.94 0.80 1.0 19.9ms
2 -8.353895362686 + -14.45 -8.92 0.80 1.0 28.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.59 0.80 1.0 17.8ms
2 -8.353895362686 + -Inf -9.49 0.80 1.0 19.7ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ │ 1.71467e-6 │ -1.7e-6 │ 2.34s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5.27783 5.27783;
5.27783 0 5.27783;
5.27783 5.27783 0]u"a₀"
Atom(Si, [ 1.31946, 1.31946, 1.31946]u"a₀")
Atom(Si, [-1.31946, -1.31946, -1.31946]u"a₀")
Since here the cell was rescaled but its shape did not change, we directly extract the optimal lattice constant from one of the cell vectors:
using AtomsBase
amin = AtomsBase.cell_vectors(results.system)[1][2]*2
println("Optimal lattice constant: ", amin)Optimal lattice constant: 10.55566292929858 a₀
Note that while for silicon the positions of the atoms are fixed by symmetry, in general a variable cell optimization will try to optimize both the cell and the positions of the individual atoms.